On Onaga´s upper bound on the mean values of probabilistic maximum flows

The reliability of capacity-limited networks subject to arc failures is evaluated by the mean value of maximum flow. Calculating the mean value of maximum flow is NP-hard. However, the Onaga upper bound sometimes gives the exact value, e.g. when graphs are bipartite. The authors give, for networks, whether arcs are directed or not, a necessary and sufficient condition for the Onaga upper bound to be exact